Pre-Algebra Examples

$| n - 3 | < 4$

Step 1

Write $| n - 3 | < 4$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$n - 3 \geq 0$

Step 1.2

Add $3$ to both sides of the inequality.

$n \geq 3$

Step 1.3

In the piece where $n - 3$ is non-negative, remove the absolute value.

$n - 3 < 4$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$n - 3 < 0$

Step 1.5

Add $3$ to both sides of the inequality.

$n < 3$

Step 1.6

In the piece where $n - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (n - 3) < 4$

Step 1.7

Write as a piecewise.

${\begin{cases} n - 3 < 4 & n \geq 3 \\ - (n - 3) < 4 & n < 3 \end{cases}$

Step 1.8

Simplify $- (n - 3) < 4$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} n - 3 < 4 & n \geq 3 \\ - n - - 3 < 4 & n < 3 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $- 3$ .

${\begin{cases} n - 3 < 4 & n \geq 3 \\ - n + 3 < 4 & n < 3 \end{cases}$

Step 2

Solve $n - 3 < 4$ when $n \geq 3$ .

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Step 2.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 2.1.1

Add $3$ to both sides of the inequality.

$n < 4 + 3$

Step 2.1.2

Add $4$ and $3$ .

$n < 7$

Step 2.2

Find the intersection of $n < 7$ and $n \geq 3$ .

$3 \leq n < 7$

Step 3

Solve $- n + 3 < 4$ when $n < 3$ .

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Step 3.1

Solve $- n + 3 < 4$ for $n$ .

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Step 3.1.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 3.1.1.1

Subtract $3$ from both sides of the inequality.

$- n < 4 - 3$

Step 3.1.1.2

Subtract $3$ from $4$ .

$- n < 1$

Step 3.1.2

Divide each term in $- n < 1$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- n < 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} > \frac{1}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} > \frac{1}{- 1}$

Step 3.1.2.2.2

Divide $n$ by $1$ .

$n > \frac{1}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $1$ by $- 1$ .

$n > - 1$

Step 3.2

Find the intersection of $n > - 1$ and $n < 3$ .

$- 1 < n < 3$

Step 4

Find the union of the solutions.

$- 1 < n < 7$

Step 5